Large Splines

This article is fundamentally about the calculations behind the ways in which computers draw graphs. In the era before computers (unknown to most of you, but very familiar to me!) graphs were drawn on paper. Typically, the data points were plotted. Next, pins were placed in the paper at these data points and a thin flexible piece of wood was threaded around these points to produce a nice smooth shape which was then traced by hand. This piece of wood was known as a spline. The word itself apparently derives from a dialect word from the East Anglia region of England for a strip of wood and is related to the word splinter. This article will present a computational method which approximates the behaviour of a spline. This method, and generalisations thereof, underlie much of computer graphics.
An earlier article in Parabola Incorporating Function (Volume 44, Number 3) showed how to construct a polynomial which passed exactly through some data points. Splines provide a different way of constructing a smooth curve which also passes exactly through the given data points and avoids some of the pitfalls which can be sometimes associated with the earlier procedure.